If a (a+b+c) = 45; b(a+b+c) = 75 and c(a+b+c) =105, then find the value of a ²+b²+c².

Correct option is D

Given:

We are given the following equations:
a(a + b + c) = 45
b(a + b + c) = 75
c(a + b + c) = 105

Formula Used:

We can solve for a, b, and c by dividing each equation by (a + b + c) and then using the values to find a² + b² + c².
a² + b² + c² can be calculated once we have a, b, and c.

Solution:

Divide each equation by (a + b + c).
​$$a = \frac{45}{a + b + c} \\\ \\b = \frac{75}{a + b + c} \\\ \\c = \frac{105}{a + b + c}$$​​

Step 2: Add the three equations to get:
​$$a + b + c = \frac{45 + 75 + 105}{a + b + c} \\\ \\\Rightarrow a + b + c = \frac{225}{a + b + c}$$​​

Let x = a + b + c.

We now have the equation:
x = 225 / x
=> x² = 225
=> x = 15

The value of (a + b + c) = 15 into the equations to find a, b, and c.
a = 45 / 15 = 3
b = 75 / 15 = 5
c = 105 / 15 = 7

Then the value of a² + b² + c².
a² + b² + c² = 3² + 5² + 7² = 9 + 25 + 49 = 83